Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, n\neq 0$. $\dfrac{{(t^{3}n^{-2})^{-5}}}{{(t^{3}n^{-5})^{4}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{3}n^{-2})^{-5} = (t^{3})^{-5}(n^{-2})^{-5}}$ On the left, we have ${t^{3}}$ to the exponent ${-5}$ . Now ${3 \times -5 = -15}$ , so ${(t^{3})^{-5} = t^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{3}n^{-2})^{-5}}}{{(t^{3}n^{-5})^{4}}} = \dfrac{{t^{-15}n^{10}}}{{t^{12}n^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-15}n^{10}}}{{t^{12}n^{-20}}} = \dfrac{{t^{-15}}}{{t^{12}}} \cdot \dfrac{{n^{10}}}{{n^{-20}}} = t^{{-15} - {12}} \cdot n^{{10} - {(-20)}} = t^{-27}n^{30}$